91Ó°ÊÓ

Evaluate the integral. \( \displaystyle \int x \tan^2 x dx \)

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \( \displaystyle \int_0^\infty \frac{x}{x^3 + 1}\ dx \)

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \( \displaystyle \int_1^\infty \frac{1 + \sin^2 x}{\sqrt{x}}\ dx \)

If \( \displaystyle \int_0^{\frac{\pi}{4}} \tan^6 x \sec x dx = I \), express the value of \( \displaystyle \int_0^{\frac{\pi}{4}} \tan^8 x \sec x dx \) in terms of \( I \).

Prove that, for even powers of sine, $$ \int_0^{\frac{\pi}{2}} \sin^{2n} x dx = \frac{1 \cdot 3 \cdot 5 \cdots \cdots (2n - 1)}{2 \cdot 4 \cdot 6 \cdots \cdots 2n} \frac{\pi}{2} $$

Evaluate the integral. \( \displaystyle \int \frac{1}{x^2 \sqrt{4x + 1}}\ dx \)

Make a substitution to express the integrand as a rational function and then evaluate the integral. \( \displaystyle \int \frac{e^x}{(e^x - 2)(e^{2x} + 1)}\ dx \)

Use integration by parts to prove the reduction formula. \( \displaystyle \int (\ln x)^n dx = x (\ln x)^n - n \displaystyle \int (\ln x)^{n - 1} dx \)

Use the Comparison Theorem to determine whether the integral is convergent or divergent. \( \displaystyle \int_1^\infty \frac{x + 1}{\sqrt{x^4 - x}}\ dx \)

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \( C = 0 \)). \( \displaystyle \int x \sin^2 (x^2) dx \)